If you have a double bowl sink in the kitchen, then the disposal is under one and the trap is under the other. It is called a "U" trap because, well, it is shaped like a "U", which means it stays filled with water so sewer gases can't come back into the house.

However above the "U" trap is a connector which routes the disposal waste to a common pipe above the U-trap. That connector isn't, as many believe, a common "T" connector, but has a baffle designed to prevent disposal waste from shooting upwards into the second sink when ejected by tyhe disposal. The downside is that the baffle reduces by half the size of the opening of the connection.

So I would disassemble that connector (as well as the U-trap) and take a look. Chances are you have some stringy celery stalk stuff or other debris clogging that baffle. When the disposal is on it "forces" water through, and water can "seep" past the clog, but only very slowly.

The piece will look like a (sideways) "T" connector above the "U" trap. Unthread it and clean it out, I'll bet that's your problem. (Snaking down through the second sink won't get it, the clog is on the othe side of the baffle, which comes from the disposal - which you can't snake.)

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